On the average of a random walk |
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Affiliation: | 1. University of Liverpool, United Kingdom;2. Xi''an Jiaotong-Liverpool University, P.R. China;3. Queen''s University Belfast, United Kingdom |
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Abstract: | Let {Sn, n ϵ N)} be a simple random walk and denote by An its time average: An = (S1+ …+Sn)/n. We give an integral test for the lower bound on An, thus giving an affirmative answer to a conjecture of P. Erdös (private communication) that An will return to a fixed region around the origin infinitely often with probability 1 in 1 dimension whereas in 2 or more dimensions it will return only finitely many times. |
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